In the theory of twinning proposed by Bilby and Crocker, the mechanism of twin formation is envisaged as a homogeneous simple shear parallel to η 1 across a plane, K 1. This phenomenological theory predicts the crystallographic forms of classical twinning modes where K 1 is an invariant plane (type I, type II, and compound). Subsequently, Bevis and Crocker also predicted non-classical modes, where the interface is not invariant and both η 1 and K 1 may be irrational. In a later treatment by Pond and Hirth, known as the topological theory, the mechanism of twin growth is modeled as inhomogeneous shear by the motion of line-defects called disconnections along the K 1 plane. For classical twins, the disconnection mechanism leads to twinning parameters consistent with phenomenological predictions, and extensive experimental evidence supporting the disconnection mechanism has been reported in the literature. By contrast, the mechanism of non-classical twin formation has not been elucidated. In the present work, we use the topological model to analyze recent experimental observations of non-classical twins in Ni2MnGa. It is demonstrated that glissile disconnections cannot form in these boundaries, so their formation must involve a novel mechanism. Here, we describe a model in which the active disconnections of two compound twins interact at their intersection, forming a misfit-free non-classical interface.

Epitaxial layers of (001) Si grown on (01T2) sapphire have been found to be misorientated by very small angles from the nominal orientation relationship. A correlation has been found between these misorientations and the degree to which vicinal substrate surfaces deviate from (01T2). This behaviour is consistent with a model where the initial surface steps are transformed into interfacial dislocations following epilayer deposition.

Deformation twins in magnesium exhibit considerable densities of I1 basal-plane stacking-faults. Since these faults generally transect their host twin, they presumably lengthen concommitantly with boundary migration during twin growth. We investigate this process using atomic-scale simulation for 101¯2 and 101¯1 twinning. It is demonstrated first that the intersection of a stacking-fault with a stationary twin boundary is delineated by a sessile imperfect disconnection. Subsequently, by applying a shear strain, we stimulate twin growth by the passage of twinning disconnections along twin boundaries, and show that these are able to propagate through such pre-existing imperfect disconnections in a conservative manner.

A model for the formation and growth of type II twins is described using the topological theory of interfacial defects and interface structures. A type I twin forms and grows to macroscopic dimensions by the generation and expansion of disconnection loops, provided these defects are sufficiently mobile. However, if their mobility is limited, they accumulate into a tilt wall, which, after accommodational relaxation, forms the type II conjugate twin. Thus, whether the type I or type II conjugate twin forms is the outcome of competitive mechanisms, depending primarily on disconnection mobility. The plausibility of this model is discussed with reference to experimental observations of twinning in alpha- U. Disconnection mobility is shown to be limited by atomic shuffling in the cases of "{176}" and "{172}" type II twins, as compared with the higher mobility expected for the active disconnections in {130} compound twins. In the topological model, the twinning shears are identical to those predicted by the classical model of deformation twinning. However, while type I twins are formed directly by shear due to the motion of disconnections on their glide planes, the mechanism of type II twinning is different, involving not only shear by disconnection motion but also accommodational relaxation. This understanding prompts a reassessment of the physical significance of the twinning elements of the classical geometrical approach. (C) 2018 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

ABSTRACT Deformation twins grow by the motion of disconnections along their interfaces, thereby coupling shear with migration. Atomic-scale simulations of this mechanism have advanced to the point where the trajectory of each atom can be followed as it transits from a site in the shrinking grain, through the interface, and onwards to a site in the growing twin. Historically, such trajectories have been factorised into shear and shuffle components according to some defined convention. In the present article, we introduce a method of factorisation consistent with disconnection motion. This procedure is illustrated for the case of twinning in hexagonal close-packed materials, and shown to agree with simulated atomic trajectories for Zr. IMPACT STATEMENT Shear and shuffle displacements accompanying () twinning are quantified consistently with growth by the observed mechanism of disconnection motion. This advance will facilitate the understanding of twinning kinetics.

At equilibrium the tensor strains and rotations produced by the network of dislocations in a semi-coherent interphase interface are equal and opposite to those arising from the coherent terraces. We express this condition using the Frank-Bilby equation. All strains and rotations are partitioned between the two phases in a manner dependent on the relative compliance of the phases. Consequently, the proposition that a parent-martensite habit plane is an invariant plane of the shape transformation is only exact in the absence of rotational distortions. Rotational distortions are inevitably produced by lattice-invariant deformation (slip or twinning), and also, generally, by transformation dislocations (disconnections).

Observations are presented, obtained by in situ straining and conventional TEM, of a transformation mechanism by coordinated secondary twinning predicted by Mullner and King. The material studied is the martensitic phase of a non-modulated Ni–Mn–Ga alloy, which exhibits a microstructure comprising domains of lamellar matrix/twin composites. Straining these specimens induced lamellar domains to transform into their conjugate counterparts. In this process, secondary twinning generates a change of misorientation between the matrix and twin lamellae of the initial domain by nearly 23°. The orientation evolves over a region behind the transformation front about 100 nm in extent.

The microstructure of a Ni–Mn–Ga alloy in the martensitic phase was investigated using transmission electron microscopy. Inter-variant twin boundaries were observed separating non-modulated tetragonal martensite variants. In addition, intra-variant boundary structures, referred to here as “conjugation boundaries”, were also observed. We propose that conjugation boundaries originate at the transformation interface between austenite and a nascent martensite variant. In the alloy studied, deformation twinning was observed, consistent with being the mode of lattice-invariant deformation, and this can occur on either of two crystallographically equivalent conjugate {101}〈101¯〉 twinning systems: conjugation boundaries separate regions within a single variant in which the active modes were distinct. The defect structure of conjugation boundaries and the low-angle of misorientation across them are revealed in detail using high-resolution microscopy. We anticipate that the mobility of such boundaries is lower than that of inter-variant boundaries, and is therefore likely to significantly affect the kinetics of deformation in the martensitic phase.

Disconnection motion along () and () twins in Zr is investigated using atomic-scale simulation. In particular, the high mobility of glissile disconnections is studied in terms of the atomic shears and shuffles involved. Using a quasi-static simulation procedure, the displacements of individual atoms are followed as they transit from matrix sites, through interfacial sites, and hence to twin sites by repeated passages of disconnections along the interface. It is found that the overall displacements for the cases studied are those predicted by the Bilby and Crocker (1965) theory which invokes homogeneous shear deformation. However, the present work enables atomic tracks to be followed through the cores of moving disconnections. The combinations of shears and shuffles in the two twinning systems are found to be quite distinct. In addition to tracking their coordinates, the variation of hydrostatic pressure experienced by the atoms is also quantified.

This chapter contains sections titled: Underlying Principles Habit Planes Reference States Summary Acknowledgement

This chapter contains sections titled: Introduction Topological Model Topological solutions (575)γ habits of lath martensite Independent transformation systems Conclusion

Li, Kadiri and Horstemeyer [1] recently studied {11 (2) over bar 2} < 11 (2) over bar (3) over bar > twinning in titanium by atomic- scale computer simulation and proposed a new mechanism in which elementary twinning dislocations (TDs) are nucleated and glide in an extended fashion on adjacent planes. In this comment, we argue that the interpretation of the simulations is in error for several reasons. First, the Burgers vector of the TDs seen in the simulations was not determined correctly. Second, these TDs do not produce the {11 (2) over bar2} twin mode known to occur in titanium. Third, the experimentally observed mode occurs under c- axis compression, whereas the motion of the twin boundary in [1] was in response to c- axis tension. The former mode cannot be simulated with the MD model used in [1]. Fourth, the temperature dependence of {11 (2) over bar2} twinning found experimentally was misunderstood in [1]. We conclude that the TD responsible for this deformation mode is the one long- established by classical twinning theory and studied at the atomic level by computer simulations performed more than 20 years ago.

The structure of disconnections in symmetrical low- and high-angle [0001] tilt boundaries in an hcp metal are studied using atomic-scale simulation. Applied engineering strains cause such defects to move conservatively along the boundaries, producing coupled shear and migration. The Peierls stresses causing such motion are found to decrease precipitously through the transition from low- to high-angle boundaries. The reason underlying this behaviour is discussed.

Li, Kadiri and Horstemeyer Citation[1] recently studied twinning in titanium by atomic-scale computer simulation and proposed a new mechanism in which elementary twinning dislocations (TDs) are nucleated and glide in an extended fashion on adjacent planes. In this comment, we argue that the interpretation of the simulations is in error for several reasons. First, the Burgers vector of the TDs seen in the simulations was not determined correctly. Second, these TDs do not produce the twin mode known to occur in titanium. Third, the experimentally observed mode occurs under c-axis compression, whereas the motion of the twin boundary in Citation[1] was in response to c-axis tension. The former mode cannot be simulated with the MD model used in Citation[1]. Fourth, the temperature dependence of twinning found experimentally was misunderstood in Citation[1]. We conclude that the TD responsible for this deformation mode is the one long-established by classical twinning theory and studied at the atomic level by computer simulations performed more than 20 years ago. Keywords: twinningdislocationtitanium Notes 1. This mapping assumes that the interface structures in Figure 3 are identical on either side of the defect. In fact, the lower crystal appears to be rigidly displaced leftwards slightly away from the mirror reflection position on the left-hand side, and displaced rightwards on the right-hand side. Such a change in the state of rigid-body displacement, if real, would modify the magnitude of b of the disconnection [6].

The physical basis for the Frank–Bilby equation is considered. Dual descriptions in terms of interface physics and mechanics are introduced. Natural (NDP), commensurate (CDP) and rotated (RCDP) dichromatic patterns are introduced. Burgers vectors are defined by symmetry operations or circuits in the CDP and RCDP. Structures are described for misfit arrays, tilt arrays, twist arrays, disconnections and combinations of these defects. The concepts of partitioning of elastic distortions, array energies, node formation, and the lateral spreading of defects within interfaces are considered. Examples with analytical solutions, numerical solutions and iterative solutions are presented. We elucidate some principles that emerge from the solutions and present reasons why some results differ from other methods of analysis.

The mechanism of coupled migration and shear is studied in a range of [0001] tilt boundaries in hexagonal close-packed metal using atomic-scale computer simulation. Symmetrical tilt boundaries spanning the low- and high-angle regimes and comprising regular arrays of grain boundary dislocations are simulated. For each misorientation, θ, the perfect boundary (pristine) is investigated as well as one containing a disconnection. Both types of structures are subjected to incremental applied strains to determine the stress that produces coupled migration and shear. The stress for motion in the pristine case, entailing nucleation, is higher than the Peierls stress for motion when disconnections are present. We conclude that the applied stresses in our simulations exert a Peach–Koehler force on pre-existing disconnections, thereby providing a feasible mechanism with a well-defined driving force that produces coupled migration and shear. This mechanism is feasible for the lower-angle boundaries studied, and facile for the high-angle cases.

Deformation twinning is investigated in the martensitic phase of a Ni46.75Mn34Ga19.25 (at.%) alloy. X-ray and electron diffraction are used to establish the crystallography of the non-modulated tetragonal martensite, and transmission electron microscopy is employed to deduce the twinning parameters. It is convenient to define the twinning parameters with respect to a “monoclinic” unit cell, designated 2M: then K1, η1, K2, and η2 are (001), [100], (100), and [001] respectively. The Burgers vector of the active twinning disconnections is close to 1/6[100] and the disconnections are associated with steps of height d(002). These defects are expected to be highly mobile since their motion does not require atomic shuffling. It is shown that periodic arrangements of two layer twins produce modulated crystal structures, such as 14M.

The 3D structures of products embedded in their parent phases are described using the topological model. Various morphologies can be formed by arrangements of concentric disconnection loops in the interfaces and dislocation loops within the product. These morphologies can exhibit extended regions of the equilibrium habit plane, but stress concentrations arise elsewhere, such as at plate edges. Some mechanisms for accommodating these stress concentrations are outlined, and supporting experimental observations are cited.